mm-309

===
Subject: Re: A valid bubble sort algorithm?Well, I'm not
familiar with Visual Basic (I think that's what you saidit
was) but it does look like Bubble Sort. In case you don't know
whatBubble Sort is, I'll give a quick explanation (and please,
someonecorrect me if I'm wrong).The program compares two side
by side values. Lower values (or highervalues, depending on
the program) will bubble to the top. Here's alittle
example.5,20,3,88,6Look at the first two terms:5,20. 5 is less
than 20, so we don't needto sort it. Then we have 20 and 3. 3
is less than 20, so it should bein front of 20.5,3,20,88,6.
Now we look at the first two, new list is 3,5,20,88,6. Welook
at the next two and then the next three. Repeat by looking at
thefirst, then second, then thrid, then fourth. You get the
idea. Thefinal result is 3,5,6,20,88 in the end.
===
Subject: Re:
Cubic Spline with mixed boundary condition >In the
construction of cubic spline(for n-knots) on a segment [a,b],
>there are 4n unknowns with 4n-2 equations. So, two more
conditions >to be put in 0order to find all the unknowns. >Let
S be the cubic spline approximation of a function f which is
four >times continuously differenetiable. >1. End conditions
of 1st type: S'(a) = f'(a), S'(b) = f'(b) the Hermite cubic
spline >2. End conditions of 2nd type: S(a) = f(a), S(b) =
f(b)you forgot the most often used one: S''(a)=0 S''(b)=0 the
natural cubic spline >There are two other conditions: Periodic
Spline(3rd type) >and not-a-Knot condition (4th type). >It is
also possible to construct spline with boundary mixed
>boundary condition from 1st type and 2nd type as S'(a) =
f'(a) >and S(a) = f(a). Is there any special name for such
type >of boundary condition ? What is the Optimal error bounds
for >the spline arising from mixed boundary condition ? this
dpends: if you look at the proves, then you see two things
entering: the local precision of the approxiamtion, obtained
from the interpolation property and the boundary condition and
the bound for the inverse, depending on the strict diagonally
dominance property. the latter will remain valid also for
mixed boundary conditions, but what about the interpolation
error if S(a)=f(a), S(a+h)=f(a+h), S''(a)=f''(a) but S'(a) not
= f'(a) ? clearly you cannot have here an O(h^4) errorbound
near a >Is the error bounds results of Hall and Meyer(
Theorem-5, Journal of >Approximation theory,16,105-122,1976)
or (Theorem 3.4,page-169, An >Introductionto Numerical
Analysis, K.E. Atkinson, 2nd edition, John >Wiley) are valid
in this case ? I don't have these sources handy, but I guess
no because ... see above Expecting favourable response from
>spline experts. >With best regards, >Bedabrata Chand
hthpeterremove nospam from my emailaddress 
===
Subject: Existence
of a unique polynomial by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i2BDngl16003;I am trying to think how do I go about figuring
out whether or notthere exists a unique polynomial of degree
four so thatf(p1)= q1; f'(p2)=q2; f'''(p3)=q3; f'''(p4)=q4 and
f''''(p5)=q5for all p1,p2,p3,p4,p5 and q1,q2,q3,q4,q5 with
p3!=p4[Notation: ' used to denote n-th differention]How do I
begin figuring out the problem? I am lacking the
thought-R
===
Subject: Re: Existence of a unique polynomial >I am
trying to think how do I go about figuring out whether or not
>there exists a unique polynomial of degree four so that
>f(p1)= q1; f'(p2)=q2; f'''(p3)=q3; f'''(p4)=q4 and
f''''(p5)=q5 >for all p1,p2,p3,p4,p5 and q1,q2,q3,q4,q5 with
p3!=p4 >[Notation: ' used to denote n-th differention] >How do
I begin figuring out the problem? I am lacking the thought >-R
nice homework.think about the fourth derivative of a fourth
order polynomial.having done this, how looks the third
derivative like and what can you conclude using p3 != p4 from
the conditions there.what remains unknown?hthpeter
===
Subject:Subject:
: Re:
Existence of a unique polynomial> I am trying to think how do I
go about figuring out whether or not> there exists a unique
polynomial of degree four so that> f(p1)= q1; f'(p2)=q2;
f'''(p3)=q3; f'''(p4)=q4 and f''''(p5)=q5> for all
p1,p2,p3,p4,p5 and q1,q2,q3,q4,q5 with p3!=p4> [Notation: '
used to denote n-th differention]> How do I begin figuring out
the problem? I am lacking the thoughtSet up a system of linear
equations and look at the determinant.Arnold Neumaier
===
Subject:
understanding time domain by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i2BDnfs15942;hi, a querywhat is the relationship with time,(t
= 0:0.001:0.6), and the rest ofthe signal or code in the
following tutorial:t = 0:0.001:0.6;x =
sin(2*pi*50*t)+sin(2*pi*120*t);y = x +
2*randn(size(t));plot(1000*t(1:50),y(1:50))title('Signal
Corrupted with Zero-Mean Random Noise')xlabel('time
(milliseconds)')It is difficult to identify the frequency
components by looking at theoriginal signal. Converting to the
frequency domain, the discreteFourier transform of the noisy
signal y is found by taking the512-point fast Fourier
transform (FFT):Y = fft(y,512);The power spectrum, a
measurement of the power at various frequencies,isPyy = Y.*
conj(Y) / 512;Graph the first 257 points (the other 255 points
are redundant) on ameaningful frequency axis: f =
1000*(0:256)/512;plot(f,Pyy(1:257))title('Frequency content of
y')xlabel('frequency (Hz)')This represents the frequency
content of y in the range from DC up toand including the
Nyquist frequency. (The signal produces the
strongpeaks.)
===
Subject: Re: Runge-Kutta and circuit simulation>
I'm trying to implement 4th-order Runge-Kutta in Matlab for a
circuit> simulator. The ODE looks something like this:> Gx +
Cx' = W> where W is the source vector and G and C are the
modified nodalmatrices> of the time invariant and reactive
components respectively.> If am interpreting my reference
material correctly then the function f> in the Runge-Kutta
method is> f := x'=C^(-1) (W - Gx)> The problem being that in
this expression C is almost certainlysingular> and as a result
it is not possible to solve for x'.> Since it is quite common
for circuit solvers to use Runge-Kuttamethods,> would someone
mind explaining what my mistake is?> If WC and GC are non
singular then isn't x(t) = -(WC)^-1 +>
exp[-(GC)^-1t]x(0)?There are typos in the forgoing: here's
what it should have beenIf C^-1W and C^-1G are non singular
(and not too ill-conditioned) thenx(t) = -C^-1W +
exp[-C^-1Gt]x(0).This is trivial to implement in Matlab using
the builtin expm.> -- > HTH,> Gerry T.
===
Subject: Free Math
HelpWe are an experienced math tutoring organisation operating
out ofCanada.To receive a free booklet from us, giving you an
example of how weteach specific math concepts, email
mathhelpbooklets@mail.com
===
Subject: constrained brownian motioni
am trying to simulate numerically the following SDE :dX = s.dt
+ n.dB (n=0.2 in practice)I am interested in the final value
of X after time 2pi. Here, s is aconstant whose value is
either +1 or -1when s= 1, i find that var( X(2.pi) ) =
0.2776when s=-1, i find that var( X(2.pi) ) = 0.2425these
values have to be compared to the theorical
valuevar=(0.2)^2.2pi=0.2513...Everything is just fine...the
points, it is reflected by considering that s changes sign.It
seems reasonnable to me that the variance of X(2pi) in the
latter caseshould be the same as in the first one. My problem
is that this intuitiondoesn't seem to be true, as i find a
variance around 0.0933, which is farfrom expected.So am i
missing something? what should be the expected value for
thevariance in the reflecting case?Sincerely
===
Subject: Re:
constrained brownian motion>i am trying to simulate
numerically the following SDE :>dX = s.dt + n.dB (n=0.2 in
practice)>I am interested in the final value of X after time
2pi. Here, s is a>constant whose value is either +1 or -1>when
s= 1, i find that var( X(2.pi) ) = 0.2776>when s=-1, i find
that var( X(2.pi) ) = 0.2425>these values have to be compared
to the theorical value>var=(0.2)^2.2pi=0.2513...Everything is
just fine...>the points, it is reflected by considering that s
changes sign.>It seems reasonnable to me that the variance of
X(2pi) in the latter case>should be the same as in the first
one. My problem is that this intuition>doesn't seem to be
true, as i find a variance around 0.0933, which is far>from
expected.>So am i missing something? what should be the
expected value for the>variance in the reflecting case?You are
missing a lot, including your specification.When x reaches a
boundary, just changing the sign ofs will not keep it from
exceeding the boundary, so Ido not know exactly what is your
model. Also, simulationmust be done much more carefully than
usual; I had a somewhat different problem 30 years ago.To see
that the variance should be reduced, replace.6 by .1. Then X
is between -.1 and .1, and so itsvariance must be less than
.02.-- This address is for information only. I do not claim
that these viewsare those of the Statistics Department or of
Purdue University.Herman Rubin, Department of Statistics,
Purdue Universityhrubin@stat.purdue.edu Phone: (765)494-6054
FAX: (765)494-0558
===
Subject: Re: constrained brownian motion>
You are missing a lot, including your specification.> When x
reaches a boundary, just changing the sign of> s will not keep
it from exceeding the boundary, so I> do not know exactly what
is your model. Also, simulation> must be done much more
carefully than usual; I had a> somewhat different problem 30
years ago.ok, i was not clear enough to describe what i am
doing. I first only changedthe sign and clamped the X value to
the boundary. Then i computed themissing random walk, i.e,
knowing that the collision occurs between t_irather at
x=0.6-eps, eps being here the difference between the
valuecomputed with s=1 and the boundary.> To see that the
variance should be reduced, replace> .6 by .1. Then X is
between -.1 and .1, and so its> variance must be less than
.02.Yep, i had noticed this strange behavior, but i was hoping
that, as theamount of noise what way smaller than the domain, i
could just forget aboutthis issue. So if you think that my
problem in not recovering the rightvariance is this issue, do
you have any idea on how to compute a realvariance in my case?
(probably the variance depends on the number ofcollisions with
the boundaries?)
===
Subject: Fortran code at the Open DirectoryI
am an editor of the Fortran source code section of the
OpenDirectory at
http://www.dmoz.org/Computers/Programming/Languages/Fortran/
Source_Code/. There are currently 160 links, to codes in the
areas includingnumerical analysis, statistics, physics,
astronomy, chemistry,engineering, XML parsing, benchmarking,
and application development.Many links were gleaned from
http://www.fortranlib.com ,http://www.fortran.com ,
andhttp://www.personal.psu.edu/faculty/h/d/hdk/fortran.html
.Please submit new links using the Suggest URL section of the
site.The Open Directory has discretion over what is added. The
link can beto a commercial product such as Numerical Recipes if
it is distributedas source code. Fortran-callable libraries can
be added to othersections.
===
Subject: Is there any book on system
of ordinary differential equations? Can somebody introduce me
some good books on how to solve system ofordinary differential
equations by finite difference? I only haveNumerical Analysis
by Richard L. Burden and J. Douglas Faires. Butthis book
doesn't tell much about system of ODEs.Peng
===
Subject: Re: Is
there any book on system of ordinary differential equations?>
Can somebody introduce me some good books on how to solve
system of>ordinary differential equations by finite
difference? I only have>Numerical Analysis by Richard L.
Burden and J. Douglas Faires. But>this book doesn't tell much
about system of ODEs.>PengGetFinite Difference Methods for
Differential Equations. 585-6 Notes for 1998. from
here:http://www.amath.washington.edu/~rjl/
booksnotes.htmlA.L.
===
Subject: eigenvalues of large, symmetric,
matrces with many zeroes.I need to compute the eigenvalues of
a symmetric,real, nXn matrix A where:1. each row of A is a
vector of zeroes with at most 4 ones2. n is very large, of
order 10^5 or 10^6.Any suggestions are greatly
appreciated.Federico Echenique
===
Subject: Re: demande> Salut>
j'ai l'honneur de vous demand.8e de m'envoiyer un programme
qui> fait l'integration en math.8ematique.> Dans l'attente
d'une r.8eponse favorable, veiller acc.8epter mes> expressions
les plus distingu.8ees.Je m'appele spasmous. J'ai dix ans.
J'habite a England. J'aime mangerles frites. Je suis alle a
l'ecole depuis quatre ans et faire lefrancais. Je ne comprands
pas rien. Salut!
===
Subject: talking tote by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i2BMx9P20733;HI. I have about 13 years experience analysing
the toteboard. For thefirst few years I learned simple things
that basically repeat but itwas only in the last 3 years that
I truely understood the odds. Therewas much more to the odds.
I would be happy to talk further aboutthis. Please email me
and let me know.
===
Subject: universal set theory with 3 valued
logicEpigone-thread: vermsmixblerOriginator:
israel@math.ubc.ca (Robert Israel)trying to eventually get
published:i've trimmed this down now to the essentials. i
submitted it to theamerican mathematical monthly but i sent
them a zipped file and theywanted pdf only. before they
replied, i spotted something i needed tochange so i changed it
and here's the version after the change.here's what happened
since last you may have checked it out:1. x∈x iff x=U.
thus, U is a hyperset and U is the onlyhyperset. this is
theorem 0 on page 7.2. the main thing is that i noticed that
if there is a universal set Uand a subsets axiom that has the
form {y∈x : A(y)} then one cantake x to be U to get
something that looks like this: {y:A(y)}. thisis discussed on
pages 4-6.3. on page 8 there is a corollary which states this:
if P(x), thepowerset of x, contains no fuzzy sets then there is
no function f thatmaps x onto P(x). (this is the contrapositive
of a theorem whichstates that if there is a function that maps
x onto P(x) then P(x)contains at least one fuzzy set.)if
anyone were to write a paper based on this, there are
twodirections i see it going. one is that with statement 2
above inmind, a bunch of other axioms then follow from the
modified subsetsaxiom. just take {y:y=y} to be the universal
set so the universal setfollows from the modified subsets
axiom. the pair set and powersetaxiom also follow, i think, as
well as perhaps others. anotherdirection is to look into class
theory and see if there is no need forclasses all together
(see last post on cardinal numbers). finally, aninvestigation
into fuzzy sets would be interesting. are there as manysets as
there are fuzzy sets? it seems to me that fuzzy sets would
berare or some such...so criticism i can use would be greatly
appreciated as i try toprepare this for submission. if you've
been published yourself, anyadvice on how to go about doing it
would also be immensely useful...
===
Subject: Re: This Week's Finds
in Mathematical Physics (Week 203)Originator:
israel@math.ubc.ca (Robert Israel)> The idea of the proof is
pretty general. Suppose we're in some category> with finite
products and countable coproducts, the former distributing >
over the latter. And, suppose we've got an object X equipped
with an > isomorphism> X = 1 + 2X (4)> so that X acts like -1.
For example, following Schanuel and Propp, > we can take the
category of sigma-polytopes and let X be the open > interval:
then isomorphism (4) says> (0,1) = (0,1/2) + {1/2} + (1/2,1)>
Or, following Houston, we can take the category of sets and
let X be > the set of finite bit-strings. Then (4) says a
finite bit-string is > either the empty bit-string, or a bit
followed by a finite bit-string. > The relation between these
two examples is puzzling to me - if anyone> understands it,
let me know! But anyway, either one works.Here's one
relationship.Consider the subset of (0,1) consisting of the
dyadic rationals,i.e. those of them form p/2^n for some
natural numbers p and n.Each such number can be uniquely
described by a finite bitstringby interpreting a 0 to mean
choose the left half of the currentinterval and 1 to mean
choose the right half of the currentinterval. At the end, take
the midpoint. For example, corresponds to 1/20 corresponds to
1/41 corresponds to 3/400 corresponds to 1/801 corresponds to
3/810 corresponds to 5/811 corresponds to 7/8etc.Dan
===
Subject:
Fixed Point Arithmetic Resources ?Originator:
israel@math.ubc.ca (Robert Israel)hi there,from past few days
i have been doing research on using Fixed Pointarithmetic
using Integers, i was able to find out some
goodresources(google) but they were related to
simpleAddition-Subtraction-Multiplication, i m looking for
some good resources ondoing Trignometric functions(sin,cos..)
through Fixed Point AtithmeticI did google advance group
search (fixed point arithmetic group:*C++*,
ORgroup:*math*....and some more) but was not able to find out
muchresources...(i also posted on alt.math... no response
yet..)I m looking for some very good resources(preferably
online) and good books(again preferably free ebooks :-) on
doing Advance Fixed Point
Arithmeticregards---------------------------------------------
----------------------experience is a good teacher..are you a
good student ??
===
Subject: Re: Fixed Point Arithmetic Resources
?Originator: israel@math.ubc.ca (Robert Israel)One classic on
the subject is Approximations for Digital Computers, byCecil
Hastings Jr. (Princeton - dating from 1955!) The ISBN is
0691079145.The basic method is to use Tchebyshev polynomial
approximation for thevarious functions on a restricted real
interval, then map the polynomiallinearly to the desired
interval. All of this can be done in integerarithmetic if you
like - just keep track of the binary point!>hi there,>from
past few days i have been doing research on using Fixed
Point>arithmetic using Integers, i was able to find out some
good>resources(google) but they were related to
simple>Addition-Subtraction-Multiplication, i m looking for
some good resources on>doing Trignometric functions(sin,cos..)
through Fixed Point Atithmetic>I did google advance group
search (fixed point arithmetic group:*C++*,
OR>group:*math*....and some more) but was not able to find out
much>resources...(i also posted on alt.math... no response
yet..)>I m looking for some very good resources(preferably
online) and good books>(again preferably free ebooks :-) on
doing Advance Fixed Point
Arithmetic>regards>-->----------------------------------------
------------------------->experience is a good teacher..>are
you a good student ??
===
Subject: Are analytic Divisors locally
algebraic?Originator: israel@math.ubc.ca (Robert Israel)Let V
be an analytic divisor in a polydysc D in C^n (i.e. V={f=0}
withf analytic)that pass through 0 (i.e. f(0)=0)Is there (up
to shrinking D) a biholomorphism G:D->D, that fixes 0 andsuch
that the image of V is algebraic (i.e. G(V)={g=0} with
gpolynomial) ?Of course the question is trivial if V is
irreducible and smooth at 0.
===
Subject: Re: Pointwise Ergodic
TheoremOriginator: israel@math.ubc.ca (Robert Israel)>Suppose
you have a map T: X -> X for which the probability measure
v>is invariant and ergodic. Also suppose v << m where m is
lebesgue>measure. Is it possible to show convergence in the
pointwise ergodic>theorem for m almost every x in X. Obviously
the ergodic theorem>itself only talks about v almost every
convergence.I don't know anything about this ergodic stuff,
but surely theanswer is clearly not?Maybe X = [0,2], T maps
[0,1) to [0,1) and maps [1,2] to [1,2],and v is concentrated
on [1,2]. Then the hypotheses say nothing whatever about how T
behaves on [0,1), so theycertainly don't imply convergence a.e.
on [0,1).[ moderator's note: Perhaps the OP meant to assume
that v is the only ergodic probability measure. But still
there would be easy counterexamples, e.g. with X = [0,2],
where T maps [0,1) to 1, v must be concentrated on [1,2], and
the behaviour of the iterates of points in [0,1) depends on
the iterates of the single point 1, which may be atypical.-
ri]>E************************
===
Subject: Re: Permutational
PolynomialOriginator: israel@math.ubc.ca (Robert Israel)> Does
anyone know a way to check if a polynomial is a permutational
polynomial> over a finite field GF(q)? I only know the naive
method, and Hermite's> criterion.I just got pointed to your
thread from a thread over inalt.math.recreational in which
we've been discussing how to find the inverseof a particular
polynomial modulo 256.I've been interested in this myself, off
and on, and have posted a fewthreads about this to various
newsgroups over the past couple years, and canshare what I've
found. Which may not be much. Understand that I'm not
amathematician, just a hobbyist. I understand group theory and
algebra, butnot to the depth and breadth that a real
mathematician does, I'm sure.The thread that received the most
interest was Invertible CubicPolynomials, modulo M at sci.math
in Nov/2000. You might want to look forthat. It is dealing
with the specific case of cubics but perhaps you'llfind
something of interest there.There's also a thread titled
Quadratic Congruential Generators in sci.mathin Sep/2001 that
might be relevant. This was looking at using quadratics
aspseudo random number generators. Doing so requires not only
that thequadratic be a polynomial but that it also have a long
cycle (usually acomplete cycle). Since it was cross-posted to a
cryptography group, a lotof the thread deals with
appropriateness for cryptography, which may beunrelated to
your interest.There's a paper by Rivest titled Permutation
Polynomials mod 2^w availableat
theory.lcs.mit.edu/~rivest/
Rivest-PermutationPolynomialsModulo2w.pswhich covers
power-of-two moduli pretty thoroughly.In one thread Bob
Silverman pointed me to H. Cohen's book A Course
inComputational Algebraic Number Theory. The context was
whether thepolynomial was inert in Q(alpha), where alpha was a
complex root of thepolynomial. At the time I had no idea what
Q(alpha) was. In the interim Ihave learnt a little about that,
but not enough. I still do not know whatinert means in this
context.In that same thread, he noted that> There is no simple
answer to this question; it is tied into questions of> higher
reciprocity laws, evaluation of the Artin map for Q(alpha),
an> effective form of the Chebotarev Density Thm, etc. etc.>
One can give a simple answer to the question when the
polynomial is> cyclotomic.> You should also look at the
Cantor-Zassenhaus algorithm for factoring a> polynomial mod p.
Basically GCD((x+s)^(p-1/2) - 1, f(x)) mod p has a 50-50>
chance of splitting f(x) for randomly chosen s. Apply this
recursively.Another idea I had is that there are polynomials
with non-integercoefficients which nevertheless have integer
outputs for integer inputs, andso ought to be considered as
candidates for permutations in GF(q).By the way, could you
give some details about Hermite's criterion as itapplies to
this problem? I'm not familiar with it, and a web search
didn'tget me any information that I could associate with this
problem.Bob H-- -- Bob Harris
=======================================================+| To
reply, carefully remove the plastic wrapper from my address
|+============================================================
Subject:========+
===
Subject: Open position for a professor at the CS
Department of ULB (Brussels, Belgium)Originator:
israel@math.ubc.ca (Robert Israel)The Faculty of Sciences of
the Free University of Brussels (ULB)announces the opening of
a full-time position for a charg.8e de cours(first
professorial level) in Computer Science, starting October
theDetails on 
===
Subject: knot theory, conway notationOriginator:
israel@math.ubc.ca (Robert Israel)I'm not sure if this is the
right news group, but i'm going to give it a shot.i'm teaching
a course in knot theory. We're using Adams' book. I decided to
go off on a tangent and play with Conway's paper describing
his notation. i'm missing something!he says that: 3- = -2 1But
if 3- is the same as 3 -1 0, then the continued fraction for
that tangle is 3/2 but the continued fraction for -2 1 is
1/2.Also, the Conway notation for 9_{48} is 21,21,21-If this
is the same as:2 1 0 + 2 1 0 + 2 1 -1 0and we connect the
northern strands and the southern strands, then that knot can
be simplified to an 8 crossing knot.seem to see what I'm doing
wrong.HELP!maegan bos
===
Subject: Re: homeomorphisms S^3 -->
S^3Originator: israel@math.ubc.ca (Robert Israel)I don't know
about homeomorphisms, but for diffeomorphisms, Hatcherproved
(the Smale conjecture) that the space of
orientation-preservingdiffeomorphisms of S^3 deformation
retracts to SO(4). In particularall orientation-preserving
diffeomorphisms of S^3 are isotopic to theidentity.> Hi all,>
it would be great if someone could help me with the following
question:> How do I find all isotopy-classes of homeomorphisms
S^3 --> S^3?> Is it true, that all homeomorphism, which
preserve orientation, are > isotopic to the identity? If not,
how can one construct a counterexample?> This question
actually comes from knot-theory, were different> definitions
on equivalence of knots exist. Two standard definitions are>
(1)isotopic knots are equivalent, (2) homeomorphic knots are
equivalent.> While the two definitions can obviously not be
equivalent in a general> 3-manifold setting, I wonder if they
are for the sphere? This seems> somewhat more nontrivial than
I expected, even though it should be a> classic question.>
References would be very nice, too...> Norman Hilbert